ABSTRACT
Abstract Deformation equation and its integrability condition (the Bianchi identity) of a nonassociative deformation in operad algebra are considered. Sabinin’s principle is reformulated in operadic terms. Key words: Operad, cohomology, deformation, Sabinin’s principle 2000 MSC: 17A01, 18D50
The Sabinin principle states that nonassociativity is an algebraic equivalent of the differential geometric concept of curvature (e.g., [1, 2]). To see the equivalence, one must represent an associator in a suitable category. In this chapter, the equivalence is clarified from the operad theoretical point of view. By using the Gerstenhaber brackets and a coboundary operator in an operad, the (formal) associator can be represented as a curvature form in differential geometry. This equation is called a deformation equation. Its integrability condition is the Bianchi identity. So the Sabinin principle can be seen in operadic terms as well: an associator is an operadic equivalent of the curvature. It is shown that the Bianchi identity does not produce any restrictions on the deformation.
